We have traced the spatial distributions of intermediate-age and old stars in nine dwarf galaxies in the distant parts of the Local Group, using multi-epoch 3.6 and 4.5 micron data from the DUST in Nearby Galaxies with Spitzer (DUSTiNGS) survey. Using complementary optical imaging from the Hubble Space Telescope, we identify the tip of the red giant branch (TRGB) in the 3.6 micron photometry, separating thermally-pulsating asymptotic giant branch (TP-AGB) stars from the larger red giant branch (RGB) populations. Unlike the constant TRGB in the I-band, at 3.6 micron the TRGB magnitude varies by ~0.7 mag, making it unreliable as a distance indicator. The intermediate-age and old stars are well mixed in two-thirds of the sample with no evidence of a gradient in the ratio of the intermediate-age to old stellar populations outside the central ~1-2'. Variable AGB stars are detected in the outer extremities of the galaxies, indicating that chemical enrichment from these dust-producing stars may occur in the outer regions of galaxies with some frequency. Theories of structure formation in dwarf galaxies must account for the lack of radial gradients in intermediate-age populations and the presence of these stars in the outer extremities of dwarfs. Finally, we identify unique features in individual galaxies, such as extended tidal features in Sex A and Sag DIG and a central concentration of AGB stars in the inner regions of NGC 185 and NGC 147.